Let \( w_{1},w_{2}\)  be two orthonormal vectors of \(\mathbb {L}^{n/2}\)  with respect \(\varphi \) , let \(W=\langle w_{1},w_{2}\rangle \)  and let \(\ell \)  be the corresponding line of \(PG_{n/2-1}(q^{2})\) . Since \(Q(x)=\varphi (x,x)\) , it is clear that \(\langle w_{i}\rangle \) , \(i=1,2\) , is a non-degenerate 2-dimensional subspace of type \(-\)  of \(\mathbb {K}^{n}\) . Then \(W\)  is a non-degenerate 4-dimensional subspace of type \(+\)  of \(\mathbb {K}^{n}\)  by Proposition 2.5.11.(ii).
If \(\ell ^{\perp }\)  denotes the \(n-3\)  subspace of \(PG_{n-1}(q)\) 
corresponding to \(W^{\perp }\) , then \(X \left( \ell ^{\perp } \right)\)  contains a
subgroup isomorphic to \(\Omega _{4}^{+}(q)\)  inducing on \(\ell ^{\perp }\)  either the identity, or a central involution, by [1]},
Proposition 4.1.6.(II). Thus \( \left|X \left( \ell ^{\perp } \right)_{p} \right|=q^{2}\) . On the
other hand, \(\left|X_{\mathcal {H}}(\ell ^{\perp }) \right|_{p}=q\)  by [1]}, Proposition 4.1.4.(II). Therefore, \(X(\ell ^{\perp }) \ne X_{\mathcal {H}}(\ell ^{\perp })\) , and hence there is \(\beta \in X(\ell ^{\perp }) - X_{\mathcal {H}}(\ell ^{\perp })\)  such that \(X_{\mathcal {H}}\left( \ell \right)\leqslant X_{\mathcal {H},\mathcal {H}^{\beta }}\)  by [1]}, Proposition 4.1.4.(II). Then \(\left[ X_{\mathcal {H}}:X_{\mathcal {H}}(\ell )\right] \)  divides \(2(q+1,n/2)(q+1,n/2-2)q^{n-3}(q^{n/2}-(-1)^{n/2})(q^{n/2-1}-(-1)^{n/2-1})2\)  again by [1]}, Proposition 4.1.4(II). So does \(\left[X_{\mathcal {H}}:X_{\mathcal {H},\mathcal {H}^{\beta }}\right] \) , as \(X_{\mathcal {H}}\left( \ell \right)\leqslant X_{\mathcal {H},\mathcal {H}^{\beta }}\) , which is the assertion.
